Cardinalities of Fuzzy Sets

Counting is one of the basic elementary mathematical activities. It comes with two complementary aspects: to determine the number of elements of a set - and to create an ordering between the objects of counting just by counting them over. For finite sets of objects these two aspects are realized by the same type of num­ bers: the natural numbers. That these complementary aspects of the counting pro­ cess may need different kinds of numbers becomes apparent if one extends the process of counting to infinite sets. As general tools to determine numbers of elements the cardinals have been created in set theory, and set theorists have in parallel created the ordinals to count over any set of objects. For both types of numbers it is not only counting they are used for, it is also the strongly related process of calculation - especially addition and, derived from it, multiplication and even exponentiation - which is based upon these numbers. For fuzzy sets the idea of counting, in both aspects, looses its naive foundation: because it is to a large extent founded upon of the idea that there is a clear distinc­ tion between those objects which have to be counted - and those ones which have to be neglected for the particular counting process.

Contenu

1. Triangular Operations and Negations (Allegro).- 1.1. Triangular Norms and Conorms.- 1.2. Negations.- 1.3. Associated Triangular Operations.- 1.4. Archimedean Triangular Operations.- 1.5. Induced Negations and Complementary Triangular Operations.- 1.6. Implications Induced by Triangular Norms.- 2. Fuzzy Sets (Andante spianato).- 2.1. The Concept of a Fuzzy Set.- 2.2. Operations on Fuzzy Sets.- 2.3. Generalized Operations.- 2.4. Other Elements of the Language of Fuzzy Sets.- 2.5. Towards Cardinalities of Fuzzy Sets.- 3. Scalar Cardinalities of Fuzzy Sets (Scherzo).- 3.1. An Axiomatic Viewpoint.- 3.2. Cardinality Patterns.- 3.3. Valuation Property and Subadditivity.- 3.4. Cartesian Product Rule and Complementarity.- 3.5. On the Fulfilment of a Group of the Properties.- 3.5.1. VAL and CART.- 3.5.2. CART and COMP.- 3.5.3. VAL and COMP.- 3.5.4. VAL, CART and COMP.- 4. Generalized Cardinals with Triangular Norms (Rondeau à la polonaise).- 4.1. Generalized FGCounts.- 4.1.1. The Corresponding Equipotency Relation.- 4.1.2. Inequalities.- 4.1.3. Arithmetical Operations.- 4.1.3.1. Addition.- 4.1.3.2. Subtraction.- 4.1.3.3. Multiplication.- 4.1.3.4. Division.- 4.1.3.5. Exponentiation.- 4.1.4. Some Derivative Concepts of Cardinality.- 4.2. Generalized FLCounts.- 4.2.1. Equipotencies and Inequalities.- 4.2.2. Addition and Other Arithmetical Operations.- 4.3. Generalized FECounts.- 4.3.1. The Height of a Generalized FECount.- 4.3.2. Singular Fuzzy Sets.- 4.3.3. Equipotencies, Inequalities and Arithmetical Questions.- List of Symbols.